Vector Algebra — Part 2: Dot product, cross product and their uses: visual lesson, CBSE Class 12 Mathematics

Lesson

Vector Algebra, Part 2

Two ways to multiply vectors: one gives a number, the other a vector.

In this lesson you will

Find the dot product and use it for angles and perpendicularity
Find the projection of one vector on another
Find the cross product and unit vectors perpendicular to two vectors
Find areas of triangles and parallelograms with the cross product

$OM = |\vec{a}|\cos\theta = \dfrac{\vec{a} \cdot \vec{b}}{|\vec{b}|}$.

Concept

What you will be able to do

Find the dot product and the angle between vectors
Test vectors for perpendicularity
Find the projection of one vector on another
Find the cross product, perpendicular unit vectors and areas

Concept

The dot (scalar) product

$\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta$ , where $\theta$ ( $0 \le \theta \le \pi$ ) is the angle between them. It is a number. $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$ .

Same unit vector: $1$. Different unit vectors: $0$.
$\cdot$	$\hat{i}$	$\hat{j}$	$\hat{k}$
$\hat{i}$	$1$	$0$	$0$
$\hat{j}$	$0$	$1$	$0$
$\hat{k}$	$0$	$0$	$1$

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